A Spatial Localizer for Constituent-Resolved Exciton Wannier Functions
The internal electron-hole structure of excitons influences optical and electric-field responses. Exciton Wannier functions have recently provided a local real-space representation of exciton bands. Because excitons are composite quasiparticles, these Wannier functions can be characterized by both their center-of-mass location and the internal electron-hole dipole. Resolving this internal structure in real space requires simultaneously locating the constituent electron and hole coordinates. However, the projected electron and hole position operators generally do not commute, so simultaneous diagonalization of these positions is impossible: their noncommutativity precludes a common eigenbasis and imposes a state-dependent lower bound on the joint spread. Existing approaches localize a single average coordinate or a specific constituent coordinate, but generally do not construct a common exciton Wannier basis that maximally localizes both constituent coordinates. Here, we construct an ``exciton spatial localizer'': a single Hermitian operator that embeds the constituent position operators within a Clifford-algebra structure and returns exciton Wannier functions, maximally localized in their electron and hole coordinates simultaneously. Our formulation is ansatz-free, gauge-invariant, and applies to the generic multiband case. In an interacting bilayer model with an isolated group of exciton bands, we show that reflection and time-reversal symmetries, or other nonsymmorphic symmetries detailed herein, enforce a pointwise traceless but nonzero quantum geometric dipole matrix, which manifests in symmetry-related pairs of exciton Wannier functions with equal and opposite internal electron-hole dipoles.